Because lattice theory is so vast, the primary purpose of this paper will be to present some of the general properties of lattices, exhibit examples of lattices, and discuss the properties of distributive and modular lattices.

Thesis (M.Sc. (Chemical Physics)) -- Mahidol University, 1979. / Financial support by the Faculty of Graduate Studies and National Research Council.

Prof. Marcel Wild completed his PhD with Zurick University and this is a copy of the original works / The original works can be found at http://www.hbz.uzh.ch/ / ABSTRACT: A linear representation of a modular lattice L is a homomorphism from L into the lattice Sub(V) of all subspaces of a vector space V. The representation theory of lattices was initiated by the Darmstadt school (Wille, Herrmann, Poguntke, et al), to large extent triggered by the linear representations of posets (Gabriel, Gelfand-Ponomarev, Nazarova, Roiter, Brenner, et al). Even though posets are more general than lattices, none of the two theories encompasses the other. In my thesis a natural type of finite lattice is identified, i.e. triangle lattices, and their linear representation theory is advanced. All of this was triggered by a more intricate setting of quadratic spaces (as opposed to mere vector spaces) and the question of how Witt’s Theorem on the congruence of finite-dimensional quadratic spaces lifts to spaces of uncountable dimensions. That issue is dealt with in the second half of the thesis.

Understanding the many-body behaviors of interacting electron systems remains one of the central topics in condensed matter physics. Novel correlated phases coupled to lattice symmetry, topological orders and hidden geometrical degrees of freedom could be induced and modulated by external electric or magnetic fields. Extensive attention have been drawn to these research directions which are of significant interests for both fundamental understanding and practical applications of many-body electron systems. In this dissertation I report optical spectroscopic studies on the Coulomb coupling, phase interplay and geometric fluctuations of interacting two-dimensional electron systems. The research provides a key approach to engineering many-body ground states and offers critical insights into their underlying nature. Electric potential or magnetic field modulations are applied to the electrons hosted in semiconductor quantum wells. Through lateral superlattice nanopatterning, we fabricate semiconductor artificial graphene where resonant inelastic light scattering is employed to characterize the engineered band structures. Flat bands hosting van Hove singularities are directly observed by optical emission. Coulomb coupling between electrons with diverging density of states are found to have significant impacts on the energies and line-shapes of the optical spectra. The results demonstrate a novel and tunable platform to explore intriguing many-body physics. External magnetic fields have been known to trigger a rich phase diagram in interacting two-dimensional electron systems, encompassing phenomena such as the fractional quantum Hall effect. The phase interplay gives rise to domain textures in the bulk of electron systems and affects the dispersion of collective excitations. We probe impacts of domain textures on low-lying neutral excitations through doubly resonant inelastic light scattering. We demonstrate that large domains of quantum fluids can support well-defined long-wavelength modes which could be interpreted by theories for uniform phases. Equipped with ultra-high mobility quantum wells and circularly polarized light scattering techniques, we resolve the spin of long-wavelength magnetoroton modes and provide characteristic evidence of the chiral graviton at Landau level filling factor $\nu= ⅓ fractional quantum Hall state. The results offer the first experimental evidence of geometrical degrees of freedom in the fractional quantum Hall effect.

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